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Quadratic number fields with class numbers divisible by a prime q

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2 Author(s)
Yang, Dong ; Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China ; Zhang, Xianke

Let q≥5 be a prime number. Let k={BBQ}(\sqrt {d}) be a quadratic number field, where d = (-1)^{(q(q-1))/(2)}\bullet (-(q-1)^{q-1}uw^{q}+u^{2}q^{q}) . Then the class number of k is divisible by q for certain integers u, w. Conversely, assume Ω/k is an unramified cyclic extension of degree q(which implies the class number of k is divisible by q), and Ω is the splitting field of some irreducible trinomial f(X) = Xq-aX-b with integer coefficients, k={BBQ}(\sqrt {D(f)}) with D(f) the discriminant of f(X). Then f(X) must be of the form f(X= Xq-uq−2 wX-uq−1 in a certain sense where u, w are certain integers. Therefore, k={BBQ}(\sqrt {d}) with d = (-1)^{(q(q-1))/(2)} (-(q-1)^{q-1}uw^{q}+u^{2}q^{q}) . Moreover, the above two results are both generalized for certain kinds of general polynomials.

Published in:

Tsinghua Science and Technology  (Volume:9 ,  Issue: 4 )